Marc Brachet

Secondary instability of a temporally growing mixing layer

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit threedimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

A simple global characterization for normal forms of singular vector fields

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

The dynamics of freely decaying two-dimensional turbulence

Direct numerical simulations of decaying high-Reynolds-number turbulence are presented at resolutions up to 800 2 for general periodic flows and 2048 2 for periodic flows with large-scale symmetries. For turbulence initially excited at large scales, we characterize a transition of the inertial energy-spectrum exponent from n ≈ − 4 at early times to n ≈ − 3 when the turbulence becomes more mature. In physical space, the first regime is associated with isolated vorticity-gradient sheets, as predicted by Saffman (1971). The second regime, which is essentially statistical, corresponds to an enstrophy cascade (Kraichnan 1967; Batchelor 1969) and reflects the formation of layers resulting from the packing of vorticity-gradient sheets. In addition to these small-scale structures, the simulation displays vorticity macro-eddies which will survive long after the vorticity-gradient layers have been dissipated (McWilliams 1984). We validate the linear description of two-dimensional turbulence suggested by Weiss (1981), which predicts that coherent vortices will survive in regions where vorticity dominates strain, while vorticity-gradient sheets will be formed in regions where strain dominates. We show that this analysis remains valid even after vorticity-gradient sheets have been formed.

An analysis of subgrid‐scale interactions in numerically simulated isotropic turbulence

Using a velocity field obtained in a direct numerical simulation of isotropic turbulence at moderate Reynolds number the subgrid‐scale energy transfer in the spectral and the physical space representation is analyzed. The subgrid‐scale transfer is found to be composed of a forward and an inverse transfer component, both being significant in dynamics of resolved scales. Energy exchanges between the resolved and unresolved scales from the vicinity of the cutoff wave number dominate the subgrid‐scale processes and the energetics of the resolved scales are unaffected by the modes with wave numbers greater than twice the cutoff wave number. Correlations between the subgrid‐scale transfer and the large‐scale properties of the velocity field are investigated.

Decaying Kolmogorov turbulence in a model of superflow

Superfluid turbulence is studied using numerical simulations of the nonlinear Schrodinger equation (NLSE), which is the correct equation of motion for superflows at low temperatures. This equation depends on two parameters: the sound velocity and the coherence length. It naturally contains nonsingular quantized vortex lines. The NLSE mass, momentum, and energy conservation relations are derived in hydrodynamic form. The total energy is decomposed into an incompressible kinetic part, and other parts that correspond to acoustic excitations. The corresponding energy spectra are defined and computed numerically in the case of the two-dimensional vortex solution. A preparation method, generating initial data reproducing the vorticity dynamics of any three-dimensional flow with Clebsch representation is given and is applied to the Taylor–Green (TG) vortex. The NLSE TG vortex is studied with resolutions up to 5123. The energetics of the flow is found to be remarkably similar to that of the viscous TG vortex. The...

Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows

Direct numerical simulations of the three‐dimensional Euler equations at resolutions up to 2563 for general periodic flows and 8643 for the symmetric Taylor–Green vortex are presented. The spontaneous emergence of flat pancakelike structures that shrink exponentially in time is observed. A simple self‐similar model that fits these observations is discussed. Focusing instabilities similar to those leading to streamwise vortices in the context of free shear layers [J. Fluid Mech. 143, 253 (1984)], are expected to subsequently concentrate the vorticity and produce isolated vortex filaments. A finite time singularity for the Euler equation is not excluded as the result of interactions among these filaments.

Numerical study of hydrodynamics using the nonlinear Schro¨dinger equation

Abstract The hydrodynamical behavior of the nonlinear Schrodinger equation is investigated by Fourier pseudo-spectral direct numerical simulations. Its dispersive and nonlinear acoustics are characterized quantitatively and an equation that describes this regime at leading order is derived. A technique that allows the preparaion of periodic initial data containing an arbitrary system of point vortices with minimal acoustic excitations is given. The Eulerian dynamics of a jet made of an array of counter rotating vortices is obtained. Sinuous and varicose instabilities are shown to take place. Finally the numerical methods best suited to study vortex-sound interactions are discussed.

Dynamo action in the Taylor–Green vortex near threshold

Dynamo action is demonstrated numerically in the forced Taylor–Green (TG) vortex made up of a confined swirling flow composed of a shear layer between two counter-rotating eddies, corresponding to a standard experimental setup in the study of turbulence. The critical magnetic Reynolds number above which the dynamo sets in depends crucially on the fundamental symmetries of the TG vortex. These symmetries can be broken by introducing a scale separation in the flow, or by letting develop a small non-symmetric perturbation which can be either kinetic and magnetic, or only magnetic. The nature of the boundary conditions for the magnetic field (either conducting or insulating) is essential in selecting the fastest growing mode; implications of these results to a planned laboratory experiment are briefly discussed.

Quantum walks in artificial electric and gravitational fields

The continuous limit of quantum walks (QWs) on the line is revisited through a new, recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations. Possible experimental realizations are also addressed.

The Taylor-Green Vortex and Fully Developed Turbulence

We here report results obtained from numerical simulations of the TaylorGreen three-dimensional vortex flow. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier-Stokes equations (with up to 2563 modes) and by power series analysis in time.The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation. The flow is smooth but has complex-space singularities within a distanceδ(t) of the real space which are manifest through an exponential tail in the energy spectrum. It is found thatδ(t) decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity (δ=0) at a finite time. These direct integration results are consistent with new presented results extending the Morf, Orszag, and Frisch temporal power series analysis from order t40 to order t80. Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis.The viscous dynamics (decay) have been studied for Reynolds numbersR (based on integral scale) up to 3000 and beyond the time tmax at which the maximum energy dissipation is achieved. Early time, highR dynamics are essentially inviscid and laminar. Then, instabilities starting at small scales, which may be driven by viscosity, make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Neartmax the small scales of the flow are nearly isotropic providedR>1000. Various features characteristic of fully developed turbulence are observed neartmax whenR=3000.